P, Q, R are three (distinct) lattice points in the plane. Prove that if (PQ + QR)^{2} < 8 area PQR + 1, then P, Q, R are vertices of a square.

**Solution**

*Easy.*

The integer coordinates imply that PQ^{2}, QR^{2}, and 2 area PQR are all integral. Hence (PQ + QR)^{2} ≤ 8 area PQR. Hence (PQ - QR)^{2} ≤ 8 area PQR - 4 PQ·QR. But 2 area PQR ≤ PQ·QR with equality if and only if PQR is a right-angle. Hence (PQ - QR)^{2} ≤ 0 with equality only if PQR is a right-angle. Hence PQ = QR and PQR is a right angle.

© John Scholes

jscholes@kalva.demon.co.uk

12 Dec 1998